Jordan property for non-linear algebraic groups and projective varieties

TL;DR

This paper extends the Jordan property to all connected algebraic groups and automorphism groups of projective varieties, showing a uniform bound depending only on dimension.

Contribution

It proves the Jordan property holds for all connected algebraic groups and automorphism groups of projective varieties, generalizing classical results.

Findings

Connected algebraic groups have the Jordan property with dimension-dependent constants.

Automorphism groups of projective varieties also possess the Jordan property.

The Jordan constant depends only on the dimension of the algebraic group.

Abstract

A century ago, Camille Jordan proved that the complex general linear group $GL_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H \le GL_n(C)$ has an abelian subgroup $H_1$ of index $[H : H_1] \le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim \, G$, and that the full automorphism group $Aut(X)$ of every projective variety $X$ has the Jordan property